Question

What is the solution to the compound inequality in interval notation?

Answer 1

Answer:

Step-by-step explanation:

Because of the word "or", this is a disjunction, or a union of all the solutions to the inequality as they present on a number line.

For the first inequality:

4(x + 1) > -4 we distribute through the parenthesis to get

4x + 4 > -4 and subtract 4 from both sides to get

4x > -8 and then divide both sides by 4 to get

x > -2.  That will be graphed on a number line with an open hole over the -2 and the arrow will go to the right where the numbers are bigger than -2. This arrow continues into positive infinity.

For the other inequality:

2x - 4 ≤ -10 we add 4 to both sides to get

2x ≤ -6 and then divide by 2 to get

x ≤ -3. That is a closed hole over -3 and the arrow goes to negative infinity (to the left). Since one arrow goes off to the right forever, and the other arrow goes off to the left forever, and the arrows do not intersect at any point, the interval notation answer is

(-∞, -3] or (-2, ∞).  The parenthesis mean that the number following it is not included (you can't include infinity, either negative or positive, because it's not actually a number) and the brackets mean that the number following it IS included. The hole is closed over the -3 so -3 is included in the solution, where the hole over the -2 is open and is not included.